Abstract
This study aimed to utilize a novel modified approach based on teaching–learning-based optimization (MTLBO), to achieve an approximate solution of optimal control problem governed by nonlinear Volterra integro-differential systems. The scheme was based upon Chebyshev wavelet and its derivative operational matrix, which eventually led to a nonlinear programming problem (NLP). The resulted NLP was solved by the MTLBO. The novel algorithm used a heuristic mechanism to intensify learning on the best students in learner phase. The new strategy was applied to improve learners’ knowledge and to structure the MTLBO. The applicability and efficiency of the MTLBO were shown for three numerical examples. The proposed algorithm was compared with the traditional TLBO algorithm and the Legendre wavelets and collocation method in the literature. The experimental results showed that the proposed MTLBO not only obtained the high-quality solutions with respect to the absolute errors but also provided results with the high speed of convergence.
Similar content being viewed by others
References
Adibi H, Assari P (2010) Chebyshev wavelet method for numerical solution of Fredholm integral equations of the first kind. Math Probl Eng 2010, Article ID 138408
Babolian E, Fattahzadeh F (2007) Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration. Appl Math Comput 188(1):417–426
Chou JH, Horng IR (1987) Optimal control of deterministic systems described by integro-differential equations via Chebyshev series. J Dyn Syst Meas Control 109(4):345–348
El-Kady M, Moussa H (2013) Monic Chebyshev approximations for solving optimal control problem with Volterra integro differential equations. Gen Math Notes 14(2):23–36
Elnegar GN (1998) Optimal control computation for integro-differential aerodynamic equations. Math Methods Appl Sci 21(7):653–664
Hosseini SG, Mohammadi F (2011) A new operational matrix of derivative for Chebyshev wavelets and its applications in solving ordinary differential equations with non-analytic solution. Appl Math Sci 5(51):2537–2548
Kalyanmoy D (2000) An efficient constraint handling method for genetic algorithms. Comput Methods Appl Mech Eng 186(2–4):311–338
Keinert F (2003) Wavelets and multiwavelets (studies in advanced mathematics). Chapman and Hall/CRC, New York
Kochetkov YUA, Tomshin VP (1978) Optimal control of deterministic systems described by integro-differential equations. Autom Remote Control 39(1):1–6
Li Y (2010) Solving a nonliear fractional differential equation using Chebyshev wavelets. Commun Nonlinear Sci Numer Simul 15(9):2284–2292
Luenberger DG, Ye Y (1984) Linear and nonlinear programming. Springer, New York
Maleknejad K, Ebrahimzadeh A (2014) Optimal control of Volterra integro-differential systems based on Legendre wavelets and collocation method. Int J Math Comput Sci 1(7):50–54
Maleknejad, K., Nosrati Sahlan, M., Ebrahimizadeh, A.: Wavelet Galerkin method for the solution of nonlinear Klein-Gordon equations by using B-spline wavelets. In: The international conference on scientific computing, Las Vegas, Nevada (2012)
Mashayekhi S, Ordokhani Y, Razzaghi M (2013) Hybrid functions approach for optimal control of systems described by integro-differential equations. Appl Math Model 37(5):3355–3368
Mohammadi F (2015) A Chebyshev wavelet operational method for solving stochastic Volterra–Fredholm integral equations. Int J Appl Math Res 4(2):217–227
Mohammadi F, Hosseini MM (2011) A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equation. J Frankl Inst 348(8):1787–1796
Rao RV, Kalyankar VD (2013) Parameter optimization of modern machining processes using teaching learning based optimization algorithm. Eng Appl Artif Intell 26(1):524–531
Rao RV, Patel V (2012) An elitist teaching learning-based optimization algorithm for solving complex constrained optimization problems. Int J Ind Eng Comput 3(4):535–560
Rao RV, Patel V (2013) Comparative performance of an elitist teaching learning based optimization algorithm for solving unconstrained optimization problems. Int J Ind Eng Comput 4(1):29–50
Rao RV, Savsani VJ, Vakharia DP (2011) Teaching learning based optimization: a novel method for constrained mechanical design optimization problems. Comput Aided Des 43(3):303–315
Rao RV, Savsani VJ, Vakharia DP (2012) Teaching learning based optimization: an optimization method for continuous non-linear large scale problem. Inf Sci 183(1):1–15
Razzaghi M, Yousefi S (2001) Legendre wavelet method for the solution of nonlinear problems in the calculus of variations. Math Comput Model 34(1–2):45–54
Saadatmandi A, Dehghan M (2010) A new operatinal matrix for solving fractional-order differential equations. Comput Math Appl 59(3):1326–1336
Tohidi E, Samadi ORN (2013) Optimal control of nonlinear Volterra integral equations via Legendre polynomials. IMA J Math Control Inf 30(1):67–83
Acknowledgements
The first author acknowledges Gonbad Kavous University, the second author appreciates the Young Researchers and Elite Club, Najafabad Branch, Islamic Azad University, Najafabad, and the third author thanks K.N. Toosi University of Technology for supporting this research work.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
Additional information
Communicated by V. Loia.
Rights and permissions
About this article
Cite this article
Khanduzi, R., Ebrahimzadeh, A. & Peyghami, M.R. A modified teaching–learning-based optimization for optimal control of Volterra integral systems. Soft Comput 22, 5889–5899 (2018). https://doi.org/10.1007/s00500-017-2933-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-017-2933-8